WASHINGTON — Outlines of a possible compromise that would more quickly deport minors arriving from Central America emerged Thursday as part of President Barack Obama’s $3.7 billion emergency border request to address the immigration crisis on the southern border.

Republicans demanded speedier deportations that the White House initially supported but dropped from its package after complaints from immigrant advocates and some Democrats. The top House and Senate Democrats pointedly left the door open to them.

“It’s not a deal-breaker,” said House Minority Leader Nancy Pelosi, D-Calif. “Let them have their face-saver. But let us have the resources to do what we have to do.”

In the Senate, Majority Leader Harry Reid, D-Nev., said: “I’m not going to block anything. Let’s see what comes to the floor.”

Reid and Pelosi made their comments after House Speaker John Boehner, R-Ohio, and Senate Republican leader Mitch McConnell, R-Ky., both said they didn’t want to give Obama a “blank check” to deal with the crisis of tens of thousands of unaccompanied children arriving at the South Texas border, many fleeing gang violence and drawn by rumors they would be able to stay in the U.S. Boehner and McConnell indicated policy changes would be necessary to win their support.

“We want to make sure we actually get the right tools to help fix the problem,” McConnell said. Obama “needs to work with us to get the right policy into effect.”

The developments came as Obama’s Homeland Security secretary, Jeh Johnson, prepared to defend the emergency spending request at a hearing of the Senate Appropriations Committee later Thursday afternoon.

At issue is the William Wilberforce Trafficking Victims Protection Reauthorization Act of 2008. Passed to give protections to sex trafficking victims, it requires court hearings for migrant youth who arrive in this country from “noncontiguous” countries — anywhere other than Mexico or Canada.

Because of enormous backlogs in the immigration court system, the result in the current crisis is that kids streaming in from El Salvador, Honduras and Guatemala are released to relatives or others in the U.S. with notices to appear at long-distant court hearings that many of them never will attend.

Republicans want the government to have the authority to treat Central American kids the same way as kids from Mexico, who can be removed quickly unless they convince Border Patrol that they have a fear of return that merits additional screening.

“I think clearly we would probably want the language similar to what we have with Mexico,” Boehner said.

White House officials have said they support such changes and indicated last week that they would be proposing them along with the emergency spending request. But advocates objected strongly, saying kids would be denied legal protections, and now the White House position is less clear.

Asked Thursday about whether the White House would consider proposals to revise the law, White House Press Secretary Josh Earnest said the administration is open to discussions about ways to better enforce it.

“That is to say, how can this law be enforced in a more efficient way?” Earnest told reporters traveling with Obama in Texas. He added that if there are others who have ideas about how to make the process more efficient, “we’re certainly open to those discussions.”

Advocates said they remained strongly opposed and expressed anger that after comprehensive immigration reform failed to advance in Congress this year, lawmakers may be headed toward a vote on deporting minors more quickly.

“They weren’t able to get immigration reform done in this Congress and this is going to be the only piece of immigration that gets done is a bill that says we’re going to deport children fleeing violence faster,” said Marielena Hincapie, executive director of the National Immigration Law Center. “If Democrats can’t stand up to this and be the party that’s protecting children and refugees, it’s a really sad day for the country.”

More than 57,000 unaccompanied children have arrived since October even as tens of thousands more have arrived traveling as families, mostly mothers with their children.

Many are trying to reunite with family members and to escape a spike in violence in their countries, but they also report hearing rumors that once here, they would be allowed to stay. Republicans blame Obama policies aimed at curbing deportations of immigrants brought into the country illegally as children for contributing to those rumors, something Obama administration officials have largely rejected.

The situation has complicated the already rancorous debate over remaking the nation’s immigration laws at a moment when Obama has declared legislative efforts to do so dead and announced plans to proceed on his own executive authority to change the flawed system where he can.

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